Candidate ranking

The output of the DFS traversal above subject to the four constrains presented is a list of candidate paths. The candidate paths in our experiments ranged from as few as 4 to as many as 196. We process the candidates through a ranking procedure to arrive at the top inferred destination of a trip. This is where the speed limits assigned by road types come into picture again. The ranking procedure makes use of information on typical speed limits along the candidate paths to find ones that best match the speed changes observed in the trip data points. We begin by first creating an ideal speed model for each candidate, then augment the model with driving behavior typically seen when making turns, and then compute a probability for the observed trip data to have been generated from the model. The candidates are ranked based on decreasing order of the probabilities. In other words,

we create a speed profile of a typical driver for each candidate path and compare it against the actual speeds observed.

Ideal speed model

The ideal speed model of a path P is a representation of the speeds that an ideal driver would follow when driving along the path under ideal conditions. An ideal driver is considered to be one who drives at exactly the speed limit, and ideal conditions imply no acceleration or decelerations in the driving speed. The model can be formally expressed as a function M of distance d and a path P . The output of such a function is the legal speed limit at distance d from the beginning of path P (assuming speed limit is same along both directions of travel).

M(d, P ) = slimit (3.1)

In a discrete representation, the ideal speed model is an array of distance and speed pairs at points where the speed limit changes along the path.

Augmenting the model

An ideal speed model can be improved by correcting the output speed in parts of the path where the vehicle would be performing a turn. Even an ideal driver in ideal conditions will decelerate to a reasonable speed to make a right or a left turn. A turn is assumed to happen exactly at the node joining the two edges that make the turn. We assume that all left turns happen at a speed of 15 mph and all right turns happen at 10 mph. The augmented model, denoted by Maug, gradually reduces the output speed to the turning

speed over a distance that depends on the acceleration and deceleration capabilities of the vehicle. Similarly, the model also incorporates the required acceleration behavior after the turn is complete. For all vehicles in this study, we use a fixed deceleration rate of 25 feet/s2

(= 7.8m/s2 _{= 0.8g, g being the acceleration of gravity), and a fixed acceleration rate of}

distance (miles) speed (mph ) 0 1 2 3 4 01 0 20 30 40 trip data ideal model augmented model

stop stop stop & right-turn stop stop

left-turn

left-turn begin

end

trip

Figure 3.3: Speed profile for a trip, along with that generated from the ideal and the augmented models for a different path.

The augmented model also incorporates the information that the vehicle must have come to a complete halt at all stop-points. Similar to the turns, the output speed is corrected around the vicinity of the stop-points as well. Figure 3.3 compares the speed values from a trip, and the values generated from the ideal speed model and the augmented model along a similar path to the same destination.

Probability of a candidate path

Given a trip T with n data points, Èti, di, siÍ; i = 1, ..., n, and a path P , we obtain the

speed values generated by the augmented model along path P at distances d1, ..., dn. We

denote these values by sÕ

1, ..., sÕn. The probability we seek is

P rËT |Maug(di, P) = s Õ

i; i = 1, ..., n È

. (3.2)

We assume independence of speed values across time and distance, which gives us the probability as n Ÿ i=1 P rË_Èti, di, siÍ|Maug(di, P) = s Õ i È . (3.3)

Therefore, for each time instant ti, we seek to compute the probability of observing

speed si when the speed should have been sÕi at distance di along the path. The probability

speed value sÕ

i, the distribution used is

f = Y _ _ ] _ _ [ N (sÕi+ sÕ_i 10,s Õ i 30) , sÕi Ø 20mph N (sÕi,1) , otherwise , (3.4)

where N (µ, ‡) signifies a Gaussian distribution with mean µ and standard deviation ‡. The distribution implies that, for speed limits of 20 mph or more, the mean driving speed is 10% higher, and 99.7% of the drivers drive between speeds of sÕ

i and s Õ i+ s

Õ

i/20. For example,

in a road with speed limit 60 mph, most drivers are assumed to drive at speeds between 60-72 mph, with 66 mph being the mean. For lower speed limits, we assume that drivers are more likely to stay close to the limit. The probability is then computed as

P rË_Èti, di, siÍ|Maug(di, P) = s Õ i È = si+‘ ˆ si≠‘ f(x)dx, (3.5)

where ‘ is a negligible number (10≠5_{). To avoid issues of precision, we take the sum of the}

logarithm of the probabilities instead of the product of the probabilities at different time instances. The ranking is not affected because of this transformation.